Osborne Reynolds.

Papers on mechanical and physical subjects (Volume 1) online

. (page 36 of 40)
Online LibraryOsborne ReynoldsPapers on mechanical and physical subjects (Volume 1) → online text (page 36 of 40)
Font size
QR-code for this ebook

the body has a higher temperature than the gas.


Thermal transpiration through an aperture in a thin plate.
111. In this case, since there is no tangential stress, we have (Art. 87)


Whence by equation (121)

. a= < 122 >-

a 2
Since p = pwe have, integrating equations (120) and (122), respectively

da VTT 2 ~

ft <>* V o^i

(**' _> / /

,- ' (1226).

aa VTT rr

r J/ r z j~ = ~ /i -"

a 4p

,3 TT

G and ^T are constants, such that -= - is the rate at which heat is


carried across a unit of area, and is the rate at which matter is carried


From equation (122 6) we have


Equation (123) can only be approximately true as a 2 is not constant;
therefore the condition [7=0 is not possible, i.e., it is only approximately
fulfilled, whence it follows that p is only approximately constant. The close-
ness of these approximations will depend on the variation of a 2 , and within
the limits of our approximation we may consider the condition to hold.

From equation (123) we see that the direction of flow of gas is opposite


to that of the flow of heat, while since a 2 oc -^ , the rate of flow of gas is

proportional to the flow of heat, to the mass of a molecule, and inversely
proportional to T, the absolute temperature.

By equation (48) we have

s dpa

pu = -j=. ,

VTT dx

or since pz 1 is constant

s da

U = -TT ( 124 )

VTT dx

which it may be noticed is of the same form as results from the equation of
transpiration through a tube when p is constant.


Thermal Impulsion.

112. In this case there is no motion, therefore

u = 0,
whence from equations (48)

This satisfies equation (120).

Substituting from equation (125) in equations (121) and (122)> and

remembering that p x = - - - , (p^U), Art. 82, we find that these

2 v TT dx

equations lead to the same result if pa? is constant.

Putting >! = ^ we have from equation (122)


d d*

whence integrating

da = _ V^# 1 6

dx 9r y r z spi

where H has the same value as in Art. 111.

fj/v* {IT*

Remembering that -=- = 1, also considering s constant, we have,


a = VTT H^ _1_ / J. _!.

x z 9 sp l r y r z \r y r z


'J/2 ""^

Also putting r y = r z , a=a when r=oo, and a = a c when r = c, and
integrating equation (126), we have


9 sp^r'
= $-(*-*), ^ (128).


In a similar way we obtain from equation (121)

*-_** *|['_I H !V da l

dx~ 7r Pl dx\(r y + rJadx\ .................. (129)

whence integrating

p l TT r v r z ) a.

and from (127)



a do?

If r be infinite p x = p = p l and -^ = 0. Therefore C^ = and

^T = ^ 52 a^ (131)

which result may be obtained directly from the value of p x> Art. 82.
From equation (128)

8 s 2 a a'

Trr 2 a


8cs 2 a,- a"

Trr 3 a.
1 da"

T i,- T i , .. a

Putting ^ = 2 a 2 and neglecting -7-

^ -^
as compared with -

* From an abstract of a paper read before the Koyal Society by Professor Maxwell, in April,
1878 (see Nature, May 9, 1878), I see that Professor Maxwell has obtained an expression for this
inequality of pressure or " stress " arising from the inequality of temperature. The result given
by Professor Maxwell is


where p. is the coefficient of viscosity, the absolute temperature, and x any one of the three
directions x, y, z. This result, when transformed to the present notation, becomes

And if we put, as in equation (80),

we have

P 1 7T T dX

It is thus seen that the two results are identical in form, but that Professor Maxwell makes
the pressure just three times as great as that given by equation (133).

In the abstract published in Nature, Maxwell has not given the details of the method by which
he arrived at his result.


/I /^\

s 2 V M V M

8 s 2
TT r 2


/^- It-


8cs*V M V M
TT r


From equation (127) we have

p*^P} = & l_Hsr y + r z ..(135)

P! 9 VTT pa r/r/

where, as before, ~ - is the quantity of heat carried across a unit of surface.
2r y r z

At points near to the surface.

113. In equations (131), (132), and (135) no account has been taken of
the discontinuity in the immediate neighbourhood of the surface ; hence the
results obtained from these equations may not hold good within the layer of
gas of thickness s, which is adjacent to the surface.

In order to take this discontinuity into account, the equations of steady
conditions should be modified in the manner described in Art. 84, but for
this particular case the same thing may be accomplished in a somewhat
simpler manner.

Suppose the solid surface to be either spherical or cylindrical at the point


considered, and put Cj for the radius. Then it is obvious that when - is


very large the pressure on the surface will be but slightly affected by the
layer immediately adjacent to the surface, i.e., putting p Ct for the pressure at
the surface, and p Cl +s f r the pressure at a distance s from the surface,

fv\ _ /y\ s*

I } ILL is small when - is large.

Pc l+ s~Pl S

When, however, the gas surrounding the surface is limited by another
surface, (which for simplicity may be taken concentric and of radius c 2 ),

then in order that 1 ' ^ mav be small, we must have 2 large as

Pc t +s-pc 3 -s s

well as .

Our equations, therefore, may be seen to hold good when the radius of


the solid surface is large compared with s, and the distance between the
opposite surfaces is also large.

On the other hand, in the limit, when either cjs or (c a c 2 )/s are
very small, p c -pi and p Cl -p c will depend entirely on the action of the gas
within the layer of thickness s immediately adjacent to the surface. In
these cases, however, when c a /s or (cj c 2 )/s are small, the action within
this layer may be easily expressed.

114. Let the temperature of the internal surface (sphere or cylinder) be
such that the mean value of a. for the molecules which rebound from this
surface (considered as a group in a uniform gas) is a Ci ; while the temperature
of the external surface is such that the mean value of a. for the molecules
which rebound is a',

The condition that d/s or (c a c 2 )/s are small, necessitates that the
molecules which come up to the inner surface arrive as from a uniform gas
such that a = a'. That is to say, none of the molecules which rebound from
the inner surface can return until their characteristics have been completely
modified by the external surface. For if (cj c 2 )/s is small, the molecules
will cross the interval between the surfaces without encounter, while if GJ/S
is small, although (d c 2 )/s may be large, the characteristics of the gas will
be but slightly affected by the internal layer at a distance s from that
surface, and, by theorem II., the approaching molecules will arrive as from
a uniform gas in the mean condition of the gas at a distance s.

I shall first consider the case in which (d - c 2 )/s is small.

The number of molecules which arrive at the inner surface is proportional
to p'af, and the number which rebound is proportional to p e a c , and since the
numbers must be the same we have

pa.' =


The momentum imparted to the surface by the incident molecules is ^

and that imparted by the rebounding molecules is ^ , therefore

( 186 >

Since the molecules which rebound from the internal surface all proceed
to the external surface, and the surfaces are concentric, we have

O. R.

C 2 2 4 v




a c/


p 2 c 2 2 a'

Equation (138) holds whatever may be the value of Cj/s provided
(c 2 cO/s is small, and it also holds when (c 2 GI)/S is large, provided d/s
is small. When c 2 /s is small and (c 2 c,)/s is large Cj may be neglected in
comparison with c 2 , and we have

Equation (139) is almost identical with what equation (132) becomes as s
approaches in value to r. If s = r, then the only difference in those two
equations is in the coefficient. In comparing these equations, however, it must
be noticed that in (132) a is not the same as a c , for a C) only refers to the one
set of molecules those which are receding from the surface, whereas a refers
to both sets.

At the surface when either cjs or (c 2 c^/'s are small

a c , + a'
a = - L 2 '

Whence making this substitution in equation (132), and putting s =r the
coefficient differs from that in equation (139) by S/TT, which shows the extent
to which discontinuity at the surface affects the result.

General equation oj impulsion.

115. From equations (132) and (139) we may form an equation which
will hold for all values of c/s.

For if the surfaces are spherical

/* : /Z

- p' _ Jl c 2 * - tf , ^ ^ , 8 ** , / Cl Cb\i V V[ ( _ t (14()>

And for cylindrical surfaces

/^- /Z

Pj^y _ ji 0,-c, ^ ^ ^ , 4 - i , /Cl c 2 \) V Jf V (1



/C C \ f C C \

where/; [-, -] and/ 6 (- , -) are respectively unity and zero when either -
\o > / \s s / g

or (c 2 -d)/s are zero, and respectively zero and unity when both d/s and
(c 2 Cj)/s are infinite.

Equations (140) and (141) have been obtained on the assumption that
the solid surfaces are either concentric spheres or concentric cylinders. But
these equations indicate what would be the difference of pressure consequent
on a difference of temperature whatever may be the shape of the surfaces,
and particularly so when d/a and (c 2 - c,)/ are finite, which are the most
important cases.


116. Comparing the equations (140) and (141) with the equation of
transpiration (101), it appears at once that when H is zero these equations
are identical in form. Hence the curves expressing the relation between the
impulsive forces and the density of the gas under any given conditions, would
be of the same character as those expressing the relation between the
inequalities of pressure and density in the case of thermal transpiration
through a particular porous plate, and it is not necessary for me again to
examine this relation.

Besides which, the experiments on impulsion, elaborate as they have
been, furnish nothing like the definite results which I have obtained in the
experiments on thermal transpiration.

117. The principal results to be deduced from experiments other than
those which are contained in this paper, are :

(1) That the force and motion are proportional to the difference of
temperature, which results are seen to follow directly from equations (124)
and (140).

(2) That with a particular instrument the forces increase with the
rarefaction up to a certain point, after which they fall off; this result also
follows directly from the equation (140).

118. Equations (124) and (140) first revealed to me the fact that the
pressure of gas at which the force would become appreciable must vary
inversely as the size of the surface.



From equation (140) it appears that up to a certain point

and since s oc - and p oc p it appears that


So that with gas at a given density the smaller the surface the greater
would be the intensity of the impulsive force ; and hence I was led to try the
fibre of silk, with which I obtained evidence of the force at densities of half
an atmosphere ; whereas in the radiometer, with vanes something like 500
times as broad as the fibre of silk, the force does not manifest itself until the
density is very small indeed.

Earlier conclusions.

119. The equations (124) and (140) show that both the forces and the
consequent motion are, cceteris paribus, proportional to the heat communicated
from the surface to the gas; for by equation (128) c a' x H where H is
proportional to the heat communicated from the surface to the gas.

The necessity of such a relation was the subject of my former paper.* I
then obtained the formula


To translate this into the symbols of the present paper


/3 VTT#

V 2 18 c 2 '


According to my intention e should have been equal , but from the


manner in which it was obtained it has the value given above (Appendix,
note 5 (&)). Hence we have

" " ' 1 e& ~ '

lo c a
Proc. Roy. Soc., 1874, p. 407.


The corresponding equation (Appendix, note 5 (a)) derived from equation
(140) is


or when - is small

and when - is large


8 1 S #

Q-7= r

y VTT c C 2

It thus appears that the present results entirely confirm the previous
results so far as they went ; and the present investigation is a completion,
not a correction, of the former one.

The present investigation shows that, besides being proportional to the
quantity of heat, the force is proportional to the linear divergence of the
lines along which the heat flows ; and hence, if these lines are parallel, no
matter how great may be the difference of temperature, the gas can exert no
pressure above the normal pressure which it will exert on all surfaces alike.
This is the case, whether the heat is communicated to gas or is spent in
causing evaporation from the surface.

The relation between the difference of pressure and the divergence of the
lines of flow affords a clear explanation of the complex phenomena of the
radiometer ; and as these phenomena have attracted a great deal of interest,
I feel that an explanation of them will not be out of place.

Divergence of the lines of flow and the radiometer.

120. We may readily obtain a graphic representation of the results
expressed by equations (124) and (140).

Let AB, fig. 12, be a plate from which heat is being communicated to
the surrounding gas. Then the lines representing the flow of heat, drawn
according to the law of conduction, are shown in the figure.




(1) The shape of these lines depends on the distribution of temperature
over AB.

Fig. 12 shows what the lines would be if AB were hot on one side and
cold on the other, the gas being at the mean temperature and of unlimited

(2) The distribution of temperature on an opposite surface, or containing
vessel, will also affect the shape of the lines of flow.

Fig. 13 shows the lines between two parallel plates opposite one another,
the inside face, H, being hotter than the opposite face, C, while the gas and
the outside faces of the plates are at the mean temperature of G and H.

Fig. 13.

(3) The shape of the lines will also depend on the shape of the hot
surface, and the nature of the surface as affecting the rate at which it com-
municates heat to the gas.




Fig. 14 shows the direction of the lines for a cup-shaped surface, supposed
to be uniformly at a higher temperature than the gas.

Fig. 14.

In all these figures the lines are supposed to be drawn so that the distance
between any two lines is somewhere between s and 2s, so that the excess of
pressure along the lines of flow depends, cceterisparibus, on the angle between
two consecutive lines. Thus the divergence of the lines indicates the excess
of pressure, the excess being, cceteris paribiis, proportional to the square of
the angle of divergence.

The shapes of the curves of flow are independent of the density of the gas,
but the distance between these lines varies inversely as the density; and
since the angle between the lines at distance s increases with s, we see that
the excess of pressure along the lines of flow increases as the density
diminishes, as long as the mean range of the molecules is not limited by the
size of the containing vessel. When this point is reached, there can be no
further increase in the mean range, and the excess of pressure will pass
through a maximum value, and then fall with the density, until the ratio of
the excess of pressure to the mean pressure becomes constant, which it will
be in the limit.

The distribution of the force of impulsion as indicated by the figures.

121. In fig. 12 the divergence of the lines of flow is much greater towards
the edges of the plates than in the centre ; hence the excess of pressure will
be greater towards the edges. In the same way, on the cold side of the plate,
the convergence of the lines of flow is greatest towards the edges, and here
the pressure will be least.

When the density of the gas is such that the width of the plate is large
compared with s, the divergence of the consecutive heat-lines at the middle
of the plate is small, which shows that there would be but little action on


this part of the plate. At the edges, however, the divergence is greater, and
there must always be action at the edges ; and the smaller the density of the
gas, or the narrower the plate, the more nearly to the middle of the plate will
the inequality of pressure extend. Thus with a very narrow plate, such as a
spider-line, we may have the inequality of pressure all over the plate, although
in the same gas, with a broad plate, the action might only extend to a distance
from the edge equal to the thickness of the spider-line.

Fig. 13 illustrates the paradox which furnished the clue to this theory.
Towards the middle of the plate the heat-lines are parallel, and consequently
the pressure would be equal and opposite on both plates, being the mean
pressure of the gas ; so that, if the plates were of unlimited extent, there
would be 110 change in the pressure on either plate due to the one being hot
and the other cold.

At the edges, however, the heat-lines diverge from the hot plate ; hence
at this point this plate would be subject to an excess of pressure, which would
tend to force the plate back against the mean pressure of the gas on the
outside. At the edges of the cold plate the heat-lines converge on to the
plate ; hence there will be a deficiency of pressure, and the tendency will be
for the pressure at the back to force the plate forward toward the hot plate.
Thus the action is not to separate the plates, but to force them both to move
in the direction of the hotter plate to cause the hot plate to run away, and
the cold plate to follow it.

Fig. 14 shows the inequality of pressure which may exist over a surface,
itself at uniform temperature, but differing from the temperature of the gas.

On the convex side the lines diverge much more rapidly than on the
concave side, and hence the inequality of pressure due to the communication
of heat will be greater on the convex side.

Stability of the equilibrium.

122. The figures give the lines of flow on the supposition that the gas is
at rest and the surfaces all rigidly fixed. The condition would then be one
of equilibrium. But in order that such a condition might be maintained, it
would be necessary that the condition should be one of stable equilibrium.
This is a point on which the foregoing reasoning furnishes us with no

It is satisfactory, therefore, to be able to see what must happen if the
equilibrium is unstable. This is shown by equation (124), which gives the
motion of the gas, so that there may be no forces.


If either the surface AB, or the containing vessel, be perfectly free to
move, then no inequality of pressure will be possible, but motion must ensue.
Equation (124) shows the law of such motion.

The. motion.

123. The motion is given by

s da.

If we suppose the containing vessel to be fixed, then, to allow of the motion
of the gas, the plate must move with the gas. On the other hand, if the
plate be held, the vessel will be carried by the gas in the opposite direction.
Such is the phenomena of the radiometer. The vanes are as nearly as
possible free to move in the vessel, so that their motion merely indicates the
motion of the gas caused by the inequality of temperature in the gas, which
inequality is maintained by the unequal temperature of the two sides of the
vanes arising from their different power of absorbing light, or, in the case of
curved vanes, by the greater temperature of the vanes as compared with the

The constraint which is put upon the vanes in a rotatory manner neces-
sarily disturbs somewhat the free motion of the gas, as must also the friction
of the pivot and other resistances. Therefore the condition of the gas within
the vessel cannot be one of absolutely equal pressure ; arid when either the
size of the vanes or the density of the gas are too great, the utmost inequality
of pressure is insufficient to overcome these resistances, and there is no
motion. If, then, exhaustion proceeds, the inequality of pressure increases,
and motion ensues the rate of which, if the vanes were absolutely free,
would increase as the density diminished, until the mean range was limited
by the size of the envelope, so that the larger the envelope the greater the
possible rate of motion. When the paths of the molecules are limited by the
size of the vessel, the motion would, if the vanes were perfectly free to move,
remain constant for all further exhaustion ; but the inequalities of pressure
which the gas is capable of exerting diminish with the further rarefaction,
and hence, in time, must cease to be sufficient to overcome the resistances to
which the motion of the vanes is subject, and then the motion ceases.

124. There are many other points about the phenomena of the radiometer,
but with most of these I have already dealt in my former papers, the reasoning
of which, so far as it goes, appears to me to be perfectly consistent with the
more complete view of the action to which I have now attained.

My chief object in introducing the phenomena of the radiometer in this


paper has been to bring out how completely impulsion belongs to the same
class of actions as thermal transpiration, and the other phenomena depending
on the relation which the size of the external objects bears to the mean range
within the gas.

The action does not depend on the distance between the hot and cold plates.

It has been supposed by some writers on the radiometer, that the action
depends essentially on the distance between the vanes and the sides of the
vessel. This distance, however, is now seen not to be of primary consequence,
as no action will result, however close the plates may be, unless they are of
limited extent of sizes comparable with the mean ranges.


125. The several steps in this investigation have now been described in
detail. They may be summarized as follows :

(1) The primary step from which all the rest may be said to follow is
the method of obtaining the equations of motion, so as to take into account
not only the normal stresses which result from the. mean motion of the
molecules at a point, but also the normal and tangential stresses which
result from a variation in the condition of the gas (assumed to be molecular).
This method is given in Sections VI., VII., and VIII.

(2) The method of adapting these equations to the case of transpiration
through tubes or porous plates is given in Section IX. The equations of
steady motion being reduced to a general equation, expressing the relation
between the rate of transpiration, the variation of pressure, the variation of
temperature, the condition of the gas, and the dimensions of the tube.

In Section X. is shown the manner in which were revealed the probable
existence (1) of the phenomena of thermal transpiration, and (2) the law of
correspondence between all the results of transpiration with different plates,
so long as the density of the gas is inversely proportional to the lateral linear
dimensions of the passage through the plate ; from which revelations originated
the idea of making experiments on thermal transpiration and transpiration
under pressure.

(3) The method of adapting the equations of steady motion to the case
of impulsion is given in Section XI.

In Section XII. is shown how it first became apparent that the extremely
low pressures at which alone the phenomena of the radiometer had been
obtained were consequent on the comparatively large size of the vanes, and


that by diminishing the size of the vanes similar results might be obtained
at higher pressures ; whence followed the idea of using the fibre of silk and

Online LibraryOsborne ReynoldsPapers on mechanical and physical subjects (Volume 1) → online text (page 36 of 40)